We need to form a group of 5 men and 2 women from:

• 7 men 👨👨👨👨👨👨👨
• 3 women 👩👩👩

Step 1: Choose 5 men from 7 👨‍💼

We select 5 men from 7. The number of ways to do this is:

(75)=7×62×1=21\binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21(57​)=2×17×6​=21

So, there are 21 ways to pick the men. ✅

Step 2: Choose 2 women from 3 👩‍💼

Now, we pick 2 women from 3:

(32)=3×22×1=3\binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3(23​)=2×13×2​=3

So, there are 3 ways to pick the women. ✅

Step 3: Multiply both choices ✖️

Since both choices are independent, we multiply them:

21×3=6321 \times 3 = 6321×3=63

🎯 Final Answer:

The group can be made in 63 ways! 🎊

Step 1: Identify the given people 👨‍💼👩‍💼

We have:

• 7 men 👨👨👨👨👨👨👨
• 3 women 👩👩👩

We need to form a group of 5 men and 2 women. ✅

Step 2: Choose 5 men from 7 👨‍💼👨‍💼👨‍💼👨‍💼👨‍💼

Since we are selecting 5 men out of 7, we use combinations:

Ways to choose 5 men=(75)=7!5!(7−5)!\text{Ways to choose 5 men} = \binom{7}{5} = \frac{7!}{5!(7-5)!}Ways to choose 5 men=(57​)=5!(7−5)!7!​ =7!5!2!=7×62×1=21= \frac{7!}{5!2!} = \frac{7 \times 6}{2 \times 1} = 21=5!2!7!​=2×17×6​=21

So, we can choose the men in 21 ways. 🏆

Step 3: Choose 2 women from 3 👩‍💼👩‍💼

Now, we select 2 women out of 3:

Ways to choose 2 women=(32)=3!2!(3−2)!\text{Ways to choose 2 women} = \binom{3}{2} = \frac{3!}{2!(3-2)!}Ways to choose 2 women=(23​)=2!(3−2)!3!​ =3!2!1!=3×22×1=3= \frac{3!}{2!1!} = \frac{3 \times 2}{2 \times 1} = 3=2!1!3!​=2×13×2​=3

So, we can choose the women in 3 ways. 🎀

Step 4: Calculate the total number of ways 🔢

Since the selection of men and women are independent, we multiply both results:

Total ways=21×3=63\text{Total ways} = 21 \times 3 = 63Total ways=21×3=63

🎯 Final Answer:

The group of 5 men and 2 women can be formed in 63 different ways! 🎊🥳